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Preheat a skillet over medium heat. Mix the butter, mayo and smoked paprika in a small bowl until well combined. Sprinkle with salt and pepper.

Lightly butter one side of each of the bread slices with the mayo mixture. Add the cream cheese to the remaining mayo mixture and mix well to combine. Spread some of the cream cheese mixture on the other side of each of the bread slices.

Distribute the Cheddar on the cream cheese-side of 4 of the bread slices. Place the remaining 4 bread slices on top. Place the sandwiches in the skillet and cook until golden on the bottom, about 3 minutes. Flip the sandwiches and place a heavy-bottomed skillet on top of the sandwiches and cook until golden brown and the cheese is melted, about 4 minutes.